Find a solution with Laplacetransform

**1. The problem statement, all variables and given/known data**

Determine the solution for

$\displaystyle y^{''}+81y=81U(t-\frac{\pi }{2})$

when $\displaystyle \left\{y(0)=12,y'(0)=18\right\}$

U(t) is the unit step function

**3. The attempt at a solution**

Laplacetransforming :

$\displaystyle s^{2}Y(s)-sy(0)-y'(0)+81Y(s)=81e^{-\frac{\pi }{2}s}$

With given data the equation becomes

$\displaystyle s^{2}Y(s)-12s-18+81Y(s)=81e^{-\frac{\pi }{2}s}$

Solving Y(s)

$\displaystyle Y(s)=81e^{-\frac{\pi }{2}s}(\frac{1}{s^{2}+9^{2}})+12(\frac{s}{{s^{2}+9 ^{2}}})+18(\frac{1}{s^{2}+9^{2}})$

Transform again:

$\displaystyle y(t)=81U(t-\frac{\pi }{2})sin(9t)+12cos9t+18sin9t$

Problem: This is wrong but I dont know what am I doing wrong..Can someone tell me what Im doing wrong? Wolframalpha.com says that the solution should be: $\displaystyle U(\frac{\pi }{2}-t)(sin(9t)-1)+sin(9t)+12cos(9t)+1$

But I dont understand how to get that answer.

Thanks!!

Re: Find a solution with Laplacetransform

Quote:

Originally Posted by

**Muffin** **1. The problem statement, all variables and given/known data**

Determine the solution for

$\displaystyle y^{''}+81y=81U(t-\frac{\pi }{2})$

when $\displaystyle \left\{y(0)=12,y'(0)=18\right\}$

U(t) is the unit step function

**3. The attempt at a solution**

Laplacetransforming :

$\displaystyle s^{2}Y(s)-sy(0)-y'(0)+81Y(s)=81e^{-\frac{\pi }{2}s}$

With given data the equation becomes

$\displaystyle s^{2}Y(s)-12s-18+81Y(s)=81e^{-\frac{\pi }{2}s}$

Solving Y(s)

$\displaystyle Y(s)=81e^{-\frac{\pi }{2}s}(\frac{1}{s^{2}+9^{2}})+12(\frac{s}{{s^{2}+9 ^{2}}})+18(\frac{1}{s^{2}+9^{2}})$

Transform again:

$\displaystyle y(t)=81U(t-\frac{\pi }{2})sin(9t)+12cos9t+18sin9t$

Problem: This is wrong but I dont know what am I doing wrong..Can someone tell me what Im doing wrong? Wolframalpha.com says that the solution should be: $\displaystyle U(\frac{\pi }{2}-t)(sin(9t)-1)+sin(9t)+12cos(9t)+1$

But I dont understand how to get that answer.

Thanks!!

Your LT of the translated unit step is wrong

$\displaystyle \mathcal{L}[u(t-\tau)]=\frac{e^{-\tau s}}{s}$

CB

Re: Find a solution with Laplacetransform

Okey thx! But still.. Something is wrong. I dont get the right answer

Re: Find a solution with Laplacetransform

Quote:

Originally Posted by

**Muffin** Okey thx! But still.. Something is wrong. I dont get the right answer

So what do you now get?

CB

Re: Find a solution with Laplacetransform

I get the same. When I inverstransform $\displaystyle \frac{81e^{\frac{-\pi }{2}}}{s}$ I get $\displaystyle 81U(t-\frac{\pi }{2})sin(9t)$

Re: Find a solution with Laplacetransform

Quote:

Originally Posted by

**Muffin** I get the same. When I inverstransform $\displaystyle \frac{81e^{\frac{-\pi }{2}}}{s}$ I get $\displaystyle 81U(t-\frac{\pi }{2})sin(9t)$

What is:

$\displaystyle \mathcal{L}^{-1}\left[ \frac{1}{s^2+a^2}\right]$ ?

That may not get you all the way to the solution but it might help.

CB